Question

Bohr model is applied to a particle of mass m and charge $q$ is moving in a plane under the influence of a transverse magnetic field (B). The energy of the charged particle in the second level will be ( h = Planck's constant)

• A. $\frac{\text{qBh}}{\pi\text{m}}$
• B. $\frac{\text{q}^2 \text{B}^2 \text{h}^2}{4\pi\text{m}}$
• C. $\frac{\text{qBh}}{2\pi\text{m}}$
• D. $\frac{2q\text{Bh}}{\pi\text{m}}$

Hint:

Combining Bohr's angular momentum quantization with magnetic force leads to discrete energy levels for charged particles.

Answer 1

The Correct Option is A

Step 1: Concept
For a particle in a magnetic field, the centripetal force is provided by the Lorentz force: $mv^2/r = qvB \implies mvr = qBr^2$. Bohr quantization: $mvr = nh/2\pi$.

Step 2: Meaning
Equating both: $qBr^2 = nh/2\pi \implies r^2 = \frac{nh}{2\pi qB}$.

Step 3: Analysis
$Energy (E) = \frac{1}{2}mv^2 = \frac{(mvr)^2}{2mr^2}$. $E = \frac{(nh/2\pi)^2}{2m(nh/2\pi qB)} = \frac{nhqB}{4\pi m}$. For $n=2$, $E = \frac{2hqB}{2\pi m} = \frac{hqB}{\pi m}$.

Step 4: Conclusion
The energy is $\frac{qBh}{\pi m}$. Final Answer: (A)